High Dimensional Functional Data Analysis

This thesis contributes to the field of high dimensional Functional Data Analysis by addressing three distinct projects. In the first project, multivariate time series curves representing joint movements are analyzed to identify differences between typically developing subjects and those with walking abnormalities like Cerebral palsy or Parkinson's disease. A single-number index is introduced, offering a stable measure of abnormality across various scenarios, that consistently aligns with the clinical gold standard the Gross Motor Function Classification System. In the second project, the Penalized Regression with Partial Differential Equation regularization framework is extended to incorporate mathematical models for linear elasticity, specifically the linear elastic PDE. This extension allows for the estimation of elastic bodies and provides additional insights into displacement, stress, and strains in the x and y directions. The applicability of this framework spans a diverse set of applications such as designing buildings, bridges, mechanical components, and medical devices. The third project introduces a novel application of penalized regression with elastic differential regularization for reconstructing a 3D object from a sampled point cloud. This is demonstrated through an application to 3D facial data. The proposed approach is then compared with established smoothing methods, including Kriging, INLA SPDE, Thin plate spline, and Penalized Regression with the Laplacian and the Diffusion Advection Reaction PDE as the penalty.